Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Metric characterizations of dimension for separable metric spaces

Authors: Ludvik Janos and Harold Martin
Journal: Proc. Amer. Math. Soc. 70 (1978), 209-212
MSC: Primary 54F45
MathSciNet review: 0474229
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A subset B of a metric space (X, d) is called a d-bisector set iff there are distinct points x and y in X with $ B = \{ z:d(x,z) = d(y,z)\} $. It is shown that if X is a separable metrizable space, then $ \dim (X) \leqslant n$ iff X has an admissible metric d for which $ \dim (B) \leqslant n - 1$ whenever B is a d-bisector set. For separable metrizable spaces, another characterization of n-dimensionality is given as well as a metric dependent characterization of zero dimensionality.

References [Enhancements On Off] (What's this?)

  • [1] L. Janos, A metric characterization of zero-dimensional spaces, Proc. Amer. Math. Soc. 31 (1972), 268-270. MR 0288739 (44:5935)
  • [2] -, Rigid subsets in Euclidean and Hilbert spaces, J. Austral. Math. Soc. 20 (1975), 66-72. MR 0428294 (55:1319)
  • [3] -, Dimension theory via bisector chains, Canad. Math. Bull. (3) 20 (1977), 313-317. MR 477795 (80j:54032)
  • [4] -, Dimension theory via reduced bisector chains (to appear).
  • [5] R. Jones and B. Scott, Metric rigidity in $ {E^n}$, Proc. Amer. Math. Soc. 53 (1975), 219-222. MR 0377824 (51:13993)
  • [6] H. Martin, Strongly rigid matrices and zero dimensionality, Proc. Amer. Math. Soc. 67 (1977), 157-161. MR 0454938 (56:13181)
  • [7] J. Nagata, Modern dimension theory, Wiley, New York, 1965. MR 0208571 (34:8380)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54F45

Retrieve articles in all journals with MSC: 54F45

Additional Information

Keywords: Bisector set, star rigid metric, strongly rigid metric
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society